DSE Quadratic Equation

A quadratic equation in one unknown \( x \) is an equation of the form

\[ax^2 + bx + c = 0,\]

where \( a \), \( b \) and \( c \) are real numbers and \( a \neq 0 \).

[DSE 2023 II Q10]

Which of the following statements about the graph of $y = 5 + (x - 3)^2$ is true?

A. The graph opens downwards.
B. The x-intercept of the graph is 3.
C. The y-intercept of the graph is 5.
D. The graph passes through the point (3, 5).

opens downwards?

As $x\to\infty$, $y\to\infty$, because

When $a>0$, the parabola opens upwards.

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When $a<0$, the parabola opens downwards.

$y = 5 + (x - 3)^2 \implies y = x^{2} - 6 x + 14$, Therefore, $a>0$ and the parabola opens upwards.

$y=0$, $(x-3^2=-5)$, therefore, the parabola does not intercept x-axis.

$x=0$, $y=5+(0-3)^2$, therefore, $y=14$ and the parabola intercepts y-axis at (0,14).

By substituting (3,5) into $y = 5 + (x - 3)^2$, $5+(3-3)^2=5$, therefore, the parabola passes through (3,5).

Therefore, D is true.